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Composition and lattice mismatch dependent dielectric constants and optical phonon modes of InAs1−x−ySbxPy quaternary alloys
Infrared Physics & Technology 67 (2014) 318–322
Contents lists available at ScienceDirect
Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Composition and lattice mismatch dependent dielectric constants and optical phonon modes of InAs1xySbxPy quaternary alloys M. Boucenna a, N. Bouarissa b,⇑, F. Mezrag b a b
Physics Department, Faculty of Science, University of M’sila, 28000 M’sila, Algeria Laboratory of Materials Physics and its Applications, University of M’sila, 28000 M’sila, Algeria
h i g h l i g h t s InAs1xySbxPy quaternary system lattice matched to InAs. Lattice mismatching InAs1xySbxPy quaternary system. Dielectric properties of InAs1xySbxPy quaternary system. Lattice dynamics of InAs1xySbxPy quaternary system.
a r t i c l e
i n f o
Article history: Received 7 May 2014 Available online 20 August 2014 Keywords: Dielectric properties Lattice dynamics Lattice matched Lattice mismatching Quaternaries
a b s t r a c t Based on the pseudopotential formalism under the virtual crystal approximation, the dielectric and lattice vibration properties of zinc-blende InAs1xySbxPy quaternary system under conditions of lattice matching and lattice mismatching to InAs substrates have been investigated. Generally, a good agreement is noticed between our results and the available experimental and theoretical data reported in the literature. The variation of all features of interest versus either the composition parameter x or the lattice mismatch percentage is found to be monotonic and almost linear. The present study provides more opportunities to get diverse high-frequency and static dielectric constants, longitudinal and transversal optical phonon modes and phonon frequency splitting by a proper choice of the composition parameters x and y (0 6 x 6 0.30, 0 6 y 6 0.69) and/or the lattice mismatch percentage. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction III–V compound semiconductors are a different class of semiconductors that have unique properties not found in the more well known silicon [1,2]. They are so named because each contains two elements, one from the third and the other from the fifth column of the periodic table. Those compounds are key materials for advanced optoelectronic devices. These devices are found in daily life such as the internet, lighting applications and disc-players. The compounds of interest exhibit the zinc-blende type crystal lattice. By using ternary or quaternary alloys of these semiconductors, the band gaps and lattice constants can be continuously varied over a wide range [2–7]. In a ternary alloy, the band gap and the lattice parameter are generally both functions of a single composition parameter. However, in quaternary alloys, there are two
⇑ Corresponding author. E-mail address: [email protected] (N. Bouarissa). http://dx.doi.org/10.1016/j.infrared.2014.08.006 1350-4495/Ó 2014 Elsevier B.V. All rights reserved.
composition parameters that allow the band gap and the lattice parameter to be selected independently, within the constraints of a given alloy-substrate system. This may give an additional degree of freedom to band gap and lattice parameter tuning as compared to ternary alloys. III–V quaternary alloy systems are potentially of great importance for many-high-speed electronic and optoelectronic devices and literature on the fundamental properties of these material systems is growing rapidly [8]. In fact the variety of material properties accessible with these materials can be employed in the design of advanced optoelectronic devices. The design and analysis of such devices as injection lasers, photodiodes, detectors, solar cells, multilayer structures, and micro-cavities requires an accurate knowledge of the optical and lattice-dynamics properties of these materials in the region near the fundamental absorption edge as well as at the higher photon energies. The quaternary alloy InAs1xySbxPy is the only quaternary with three group-V elements that has been studied in the literature [2]. Lattice matched to InAs, it is a promising material for the
M. Boucenna et al. / Infrared Physics & Technology 67 (2014) 318–322
production of infrared light sources for the detection of gases in the 2–4 lm region of the spectrum [8–10]. In spite of the importance of InAs1xySbxPy quaternary system, its fundamental properties, such as optical constants and lattice vibration modes and their dependence on the x and y alloy compositions have not been fully explored. In the present contribution, the optical and vibrational properties of the quaternary alloy InAs1xySbxPy lattice matched to InAs are reported. Special attention is given to the investigation of the properties of interest when the lattice matching condition is not satisfied (lattice mismatched alloys). The calculations are mainly based on the pseudopotential formalism. Subsequent studies derived a vanishing bowing parameter for the quaternary alloy under load [9,10]. Moreover, Vurgaftman et al. [2] have recommended a band gap bowing parameter for the quaternary system of interest equals to zero. Thus, for the treatment of the band parameters of the material under consideration, we have used the virtual crystal approximation (VCA) in which the atomic potential is replaced by a weighted average of those of the individual elements. 2. Computational method The calculations are mainly based on the empirical pseudopotential method (EPM) within the VCA. To calculate the electronic band structure it is necessary to solve the Schrödinger equation, for which the one electron pseudopotential Hamiltonian can be written as: 2
H ¼ ðh =2mÞr2 þ V p ðrÞ
319
quaternary system InAs1xySbxPy being studied here, VS and VA are expressed as: InAS V alloy ¼ xV InSb þ yV InP S S þ ð1 x yÞV S S
ð2Þ
InAS V alloy ¼ xV InSb þ yV InP A A þ ð1 x yÞV A A
ð3Þ
The variation of the lattice constant of the quaternary alloy under load as a function of compositions x and y is determined as:
aalloy ¼ xaInSb þ yaInP þ ð1 x yÞaInAS
ð4Þ
where aInSb, aInP and aInAs are the lattice constants of InSb, InP and InAs, respectively. The longitudinal optical (LO) and transversal optical (TO) phonon frequencies have been determined by combining the Lyddane–Sachs–Teller relation [13] with the expression [14]: 2
x2LO x2TO ¼
4p eT e2 M X0 e1
ð5Þ
where e1, eT , M and X0 are the high-frequency dielectric constant, the transverse effective charge, twice the reduced mass and the volume occupied by one atom, respectively. eT , e1 and e0 have been calculated using the same procedure as that used by Bouarissa et al. [15], where the refractive index is determined using Hervé and Vandamme relation [16], e1 is taken to be the square of the refractive index, and e0 is obtained within the Harrison model as reported in [14]. 3. Results and discussion
ð1Þ
where Vp(r) is the smoothly-varying pseudopotential which can be expanded into a Fourier series over the reciprocal lattice. Because the pseudopotential in a crystal lattice is periodic, it follows that the pseudo-wave function corresponding to (1) is also periodic. The valence electrons are assumed to be weakly bound to their host atoms so that the pseudo-wave function can be expanded into a Fourier series of plane waves. The EPM involves adjusting the pseudopotential form factors so as to reproduce measured band-gap energies at selected points in the Brillouin zone. The empirical pseudopotential parameters are optimized using the non-linear least-squares method [11,12]. The experimental band-gap energies at room temperature for InAs, InSb and InP at U, X and L high-symmetry points used in the fitting procedure are given in Table 1. The solution to the energy eigenvalues and corresponding eigenvectors can then be found by diagonalizing the Hamiltonian matrix. For a typical semiconductor system, 136 plane waves are sufficient, each corresponding to vectors in the reciprocal lattice, to expand the pseudopotential. The final local adjusted symmetric VS and antisymmetric VA pseudopotential form factors and the used lattice constants at room temperature for InAs, InSb and InP compounds are listed in Table 2. The formalism can be easily generalized to the case of an alloy using the VCA. In practice, VS and VA of an alloy can be expressed in terms of those of the binary compounds involved. Hence, for the
Table 1 Experimental band-gap energies at room temperature [3] for InAs, InSb and InP fixed in the fits. Compound
EC C (eV)
EXC (eV)
ELC (eV)
InAs InSb InP
0.36 0.18 1.35
1.37 1.63 2.21
1.07 0.93 2.05
The lattice matching condition for InAs1xySbxPy on the InAs substrate is as follows:
y ¼ 2:2857143x
ð0 6 x 6 0:30; 0 6 y 6 0:69Þ
ð6Þ
The dielectric constant is a measure of a substance’s ability to insulate charges from each other. Thus, it is a material’s ability to store a charge when used as a capacitor dielectric. In the present work, both dielectric constants, namely e1 and e0 have been calculated for different composition parameters x and y. Our results regarding e1 and e0 for InAs, InSb and InP are listed in Table 3. Also shown for comparison are the available data reported in the literature. Note that for both quantities e1 and e0, our results agree generally well with those reported in Refs. [13,17,18]. The variation of e1 and e0 as a function of composition x for InAs1xySbxPy lattice matched to InAs substrate is displayed in Fig. 1. We observe that as the Sb content increases from 0 to 0.30, e1 decreases from 14.08 to 11.12. The same behavior can be also seen for e0 which decreases from 15.41 to 13.73. The decrease of the dielectric constants by increasing the composition x means that the polarity becomes lower and the ability to stabilize charges for the material of interest becomes weaker. When the lattice matching condition given by Eq. (6) is not satisfied, for each couple (x, y) (0 6 x 6 0.30, 0 6 y 6 0.69), there is of course a difference between the lattice constant of the quaternary alloy of interest and that of the substrate (InAs in this work). The lattice mismatch Da/aInAs is expressed by:
ðaalloy aInAs Þ Da ¼ 100% aInAs aInAs
ð7Þ
where aInAs is the lattice constant of InAs given in Table 2. Fig. 2 illustrates the lattice mismatch dependence of e1 and e0 for InAs0.9ySb0.1Py. Note that as the lattice mismatch is varied on going from Da/aInAs = 2% to Da/aInAs = 0.7%, both parameters e1 and e0 are highly increased. This suggests the increase of the ability of the quaternary system under load to hold large quantities of charge for a long period of time.
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Table 2 Pseudopotential parameters for InAs, InSb and InP. Compound
Form factors (Ry)
InAs InSb InP
Lattice constant (Å)
VS(3)
VS(8)
VS(11)
VA(3)
VA(4)
VA(11)
0.217669 0.201294 0.213870
0.011 0.01 0.00
0.041633 0.028338 0.070502
0.054731 0.064495 0.088816
0.039485 0.03 0.06
0.024 0.015 0.03
Table 3 Optical high-frequency dielectric constant (e1) and optical static dielectric constant (e0) of InAs, InSb and InP.
16
15 Compound
e1
e0
InAs
14.08a; 11.6b 12.3c; 12.3d
15.42a; 14.3b 15.15c; 14.9d
14
InSb
15.43a; 15.3b 15.7c; 15.6d
17.99a; 17.2b 16.8c; 17.7d
13
a
InP a b c d
b
a
9.20 ; 9.9 9.61c; 9.6d
11.85 ; 12.9 12.5c; 12.4d
6.058 6.49 5.869
InAs0.9-y Sb0.1 Py
b
12
This work. Ref. [17]. Ref. [18]. Ref. [13].
11
10
High frequency dielectric constant Static dielectric constant
9
16
-2.0
InAs1-x-y Sbx Py /InAs
15
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 2. High-frequency and static dielectric constants in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
14 Table 4 Longitudinal optical (LO) and transversal optical (TO) phonon frequencies of InAs, InSb and InP.
13
Compound
12
11
10 0.00
InAs InSb InP
High frequency dielectric constant Static dielectric constant 0.05
0.10
0.15
0.20
a b
0.25
0.30
Composition X Fig. 1. High-frequency and static dielectric constants in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
Phonons play a major role in many of the physical properties of condensed matter, such as thermal conductivity and electrical conductivity. Their accurate description results in better understanding of structural parameters responsible for the efficiency of optical devices [19,20]. In the zinc-blende structures, such as the case here, one has to deal only with the transverse optical (TO) phonon frequency (xTO) and the longitudinal optical (LO) phonon frequency (xLO) at the U-point in the Brillouin zone. In the present contribution, xTO and xLO have been calculated at the zone-center for zinc-blende InAs1xySbxPy system. Our results for InAs, InSb and InP are given in Table 4. Also shown for comparison are the available experimental and theoretical data in the literature. The agreement between our results and data quoted in Ref. [13] seems to be reasonably good. The variation in the TO and LO phonon frequencies for zincblende InAs1xySbxPy lattice matched to InAs as a function of the composition x in the range 0–0.30 is shown in Fig. 3. We observe that by increasing Sb concentration x, both mode frequencies, i.e. xTO
xLO (1013 s1) a
b
5.20 ; 4.5 3.42a; 3.7b 7.1a; 6.5b
xTO (1013 s1) 4.97a; 4.1b 3.16a; 3.5b 6.25a; 5.7b
This work. Ref. [13].
and xLO shift to high frequencies. The shift is monotonic. Thus, larger values of xTO and xLO can be obtained at large compositions x. In Fig. 4, xTO and xLO are plotted against the lattice mismatch for InAs0.9ySb0.1Py. Note that by varying the lattice mismatch from 2% to 0.7%, both parameters of interest, namely xTO and xLO decrease monotonically and are shifted to low frequencies. The trend of the TO and LO frequencies with respect to x in Fig. 3 reflects the widening of the LO–TO splitting as one proceeds from x = 0 to x = 0.30. In this respect, the LOTO splitting in InAs1xySbxPy lattice matched to InAs has been plotted as a function of Sb content (x) in Fig. 5. An inspection of Fig. 5. shows that as the Sb concentration increases from 0 to 0.30, the LO–TO splitting increases as well from 0.23 1013 to 0.67 1013 s1. The increase is monotonic and almost linear. On the other hand, as can be seen from Fig. 4, the trend of the TO and LO frequencies with respect to the lattice mismatch reflects a the narrowing of the LO–TO splitting as one goes from aDInAs ¼ 2% a to aDInAs ¼ 0:7%. In this regard, the LO–TO splitting in InAs0.9ySb0.1Py is plotted versus the lattice mismatch in Fig. 6. We observe that as the lattice mismatch varies from 2% to 0.7%, the LO–TO splitting decreases monotonically and almost linearly from 0.78 1013 to 0.24 1013 s1. It is worth noting that for all compositions and lattice mismatching of interest for the quaternary system in question, the LO–TO splitting at the U high-symmetry point in
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M. Boucenna et al. / Infrared Physics & Technology 67 (2014) 318–322
6.25
6.00
InAs1-x-y Sbx Py /InAs 5.75
5.50
5.25
5.00
TO phonon frequency LO phonon frequency
4.75
4.50 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Composition X Fig. 3. Frequencies of the LO and TO phonons in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
7.0
TO phonon frequency LO phonon frequency
6.5
6.0
5.5
5.0
InAs0.9-y Sb0.1 Py 4.5 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 4. Frequencies of the LO and TO phonons in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
0.8
0.7
InAs1-x-y Sbx Py /InAs 0.6
0.5
0.4
0.3
0.2
0.1 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Composition X Fig. 5. LO–TO splitting in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
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M. Boucenna et al. / Infrared Physics & Technology 67 (2014) 318–322
0.9 0.8
InAs0.9-y Sb0.1 Py
0.7 0.6 0.5 0.4 0.3 0.2 0.1 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 6. LO–TO splitting in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
the Brillouin zone does not vanish. This could be due to the partial ionic bonding of the material system under load. 4. Conclusion In summary, the dielectric and vibrational properties of lattice matched and mismatched InAs1x-ySbxPy quaternary alloys have been investigated within the framework of the EPM under the VCA. The agreement between our findings and the available experimental and theoretical data reported in the literature is found to be reasonably good. The composition x dependence of dielectric constants, optical phonon frequencies and LO–TO splitting for InAs1xySbxPy quaternary system lattice matched to InAs has been studied showing a monotonic behavior of all parameters of interest versus x (0 6 x 6 0.30). An inspection of the variation of the static and high-frequency dielectric constants, LO and TO phonon frequencies and LO–TO splitting as a function of lattice mismatch percentage in InAs0.9ySb0.1Py showed that all quantities of interest are strongly dependent on the lattice mismatch percentage. The results obtained in the present contribution suggest that the variation of the alloy composition x and/or the lattice mismatch percentage of the quaternary system of interest may lead to more opportunities for obtaining diverse optical and lattice dynamical properties, which can provide a basis for understanding future device concepts and applications. Conflict of interest There is no conflict of interest. References [1] S. Adachi, Physical Properties of III–V Semiconductor Compounds, Wiley, New York, 1992.
[2] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, Band parameters for III–V compound semiconductors and their alloys, J. Appl. Phys. 89 (2001) 5815– 5875. [3] S. Adachi, Band gaps and refractive indices of AlGaAsSb, GaInAsSb, and InPAsSb: key properties for a variety of the 2–4 lm optoelectronic device applications, J. Appl. Phys. 61 (1987) 4869–4876. and references therein. [4] N. Bouarissa, Effects of compositional disorder upon electronic and lattice properties of GaxIn1xAs, Phys. Lett. A 245 (1998) 285–291. [5] N. Bouarissa, The effect of compositional disorder on electronic band structure in GaxIn1xAsySb1y alloys lattice matched to GaSb, Superlattices Microstruct. 26 (1999) 279–287. [6] M. Boucenna, N. Bouarissa, Energy gaps and lattice dynamic properties of InAsxSb1x, Mater. Sci. Eng. B138 (2007) 228–234. [7] S. Adachi, Springer Handbook of Electronic and Photonic Materials, Springer, US, 2007, pp. 735–752. [8] S. Adachi, Properties of Semiconductor Alloys: Group-IV, III–V and II–VI Semiconductors, John Wiley & Sons Ltd., 2009, ISBN 978-0-470-74369-0. [9] H. Mani, E. Tournié, J.L. Lazzari, C. Alibert, A. Joullié, B. Lambert, Liquid phase epitaxy and characterization of InAs1xySbxPy on (1 0 0) InAs, J. Cryst. Growth 121 (1992) 463–472. [10] L.-C. Chen, W.-J. Ho, M.-C. Wu, Growth and characterization of high-quality InAs0.86Sb0.05P0.09 alloy by liquid-phase epitaxy, Jpn. J. Appl. Phys. 38 (1999) 1314–1316. [11] T. Kobayasi, H. Nara, Properties of nonlocal pseudopotentials of Si and Ge optimized under full interdependence among potential parameters, Bull. Coll. Med. Sci. Tohoku Univ. 2 (1993) 7–16. [12] A. Bechiri, N. Bouarissa, Energy band gaps for GaxIn1xAsyP1y alloys lattice matched to different substrates, Superlattices Microstruct. 39 (2006) 478–488. [13] C. Kittel, Introduction to Solid-State Physics, fifth ed., Wiley, New York, 1976. [14] S. Yu Davydov, S.K. Tikhonov, Pressure dependence of the dielectric and optical properties of wide-gap semiconductors, Semiconductors 32 (1998) 947–949. [15] N. Bouarissa, S. Bougouffa, A. Kamli, Energy gaps and optical phonon frequencies in InP1xSbx, Semicond. Sci. Technol. 20 (2005) 265–270. [16] P. Hervé, L.K.J. Vandamme, General relation between refractive index and energy gap in semiconductors, Infrared Phys. Technol. 35 (1994) 609–615. [17] S. Adachi, Properties of Group IV III–V and II–VI Semiconductors, Wiley, Chichester, 2005. [18] M. Levinshtein, S. Rumyantsev, M. Shur (Eds.), Handbook Series on Semiconductor Parameters, vol. 2, World Scientific, Singapore, 1999. [19] S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73 (2001) 515–562. [20] N. Bouarissa, S. Saib, Ab initio study of lattice vibration and polaron properties in zinc-blende AlxGa1xN alloys, J. Appl. Phys. 108 (2010). 113710-113710-6.
Contents lists available at ScienceDirect
Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Composition and lattice mismatch dependent dielectric constants and optical phonon modes of InAs1xySbxPy quaternary alloys M. Boucenna a, N. Bouarissa b,⇑, F. Mezrag b a b
Physics Department, Faculty of Science, University of M’sila, 28000 M’sila, Algeria Laboratory of Materials Physics and its Applications, University of M’sila, 28000 M’sila, Algeria
h i g h l i g h t s InAs1xySbxPy quaternary system lattice matched to InAs. Lattice mismatching InAs1xySbxPy quaternary system. Dielectric properties of InAs1xySbxPy quaternary system. Lattice dynamics of InAs1xySbxPy quaternary system.
a r t i c l e
i n f o
Article history: Received 7 May 2014 Available online 20 August 2014 Keywords: Dielectric properties Lattice dynamics Lattice matched Lattice mismatching Quaternaries
a b s t r a c t Based on the pseudopotential formalism under the virtual crystal approximation, the dielectric and lattice vibration properties of zinc-blende InAs1xySbxPy quaternary system under conditions of lattice matching and lattice mismatching to InAs substrates have been investigated. Generally, a good agreement is noticed between our results and the available experimental and theoretical data reported in the literature. The variation of all features of interest versus either the composition parameter x or the lattice mismatch percentage is found to be monotonic and almost linear. The present study provides more opportunities to get diverse high-frequency and static dielectric constants, longitudinal and transversal optical phonon modes and phonon frequency splitting by a proper choice of the composition parameters x and y (0 6 x 6 0.30, 0 6 y 6 0.69) and/or the lattice mismatch percentage. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction III–V compound semiconductors are a different class of semiconductors that have unique properties not found in the more well known silicon [1,2]. They are so named because each contains two elements, one from the third and the other from the fifth column of the periodic table. Those compounds are key materials for advanced optoelectronic devices. These devices are found in daily life such as the internet, lighting applications and disc-players. The compounds of interest exhibit the zinc-blende type crystal lattice. By using ternary or quaternary alloys of these semiconductors, the band gaps and lattice constants can be continuously varied over a wide range [2–7]. In a ternary alloy, the band gap and the lattice parameter are generally both functions of a single composition parameter. However, in quaternary alloys, there are two
⇑ Corresponding author. E-mail address: [email protected] (N. Bouarissa). http://dx.doi.org/10.1016/j.infrared.2014.08.006 1350-4495/Ó 2014 Elsevier B.V. All rights reserved.
composition parameters that allow the band gap and the lattice parameter to be selected independently, within the constraints of a given alloy-substrate system. This may give an additional degree of freedom to band gap and lattice parameter tuning as compared to ternary alloys. III–V quaternary alloy systems are potentially of great importance for many-high-speed electronic and optoelectronic devices and literature on the fundamental properties of these material systems is growing rapidly [8]. In fact the variety of material properties accessible with these materials can be employed in the design of advanced optoelectronic devices. The design and analysis of such devices as injection lasers, photodiodes, detectors, solar cells, multilayer structures, and micro-cavities requires an accurate knowledge of the optical and lattice-dynamics properties of these materials in the region near the fundamental absorption edge as well as at the higher photon energies. The quaternary alloy InAs1xySbxPy is the only quaternary with three group-V elements that has been studied in the literature [2]. Lattice matched to InAs, it is a promising material for the
M. Boucenna et al. / Infrared Physics & Technology 67 (2014) 318–322
production of infrared light sources for the detection of gases in the 2–4 lm region of the spectrum [8–10]. In spite of the importance of InAs1xySbxPy quaternary system, its fundamental properties, such as optical constants and lattice vibration modes and their dependence on the x and y alloy compositions have not been fully explored. In the present contribution, the optical and vibrational properties of the quaternary alloy InAs1xySbxPy lattice matched to InAs are reported. Special attention is given to the investigation of the properties of interest when the lattice matching condition is not satisfied (lattice mismatched alloys). The calculations are mainly based on the pseudopotential formalism. Subsequent studies derived a vanishing bowing parameter for the quaternary alloy under load [9,10]. Moreover, Vurgaftman et al. [2] have recommended a band gap bowing parameter for the quaternary system of interest equals to zero. Thus, for the treatment of the band parameters of the material under consideration, we have used the virtual crystal approximation (VCA) in which the atomic potential is replaced by a weighted average of those of the individual elements. 2. Computational method The calculations are mainly based on the empirical pseudopotential method (EPM) within the VCA. To calculate the electronic band structure it is necessary to solve the Schrödinger equation, for which the one electron pseudopotential Hamiltonian can be written as: 2
H ¼ ðh =2mÞr2 þ V p ðrÞ
319
quaternary system InAs1xySbxPy being studied here, VS and VA are expressed as: InAS V alloy ¼ xV InSb þ yV InP S S þ ð1 x yÞV S S
ð2Þ
InAS V alloy ¼ xV InSb þ yV InP A A þ ð1 x yÞV A A
ð3Þ
The variation of the lattice constant of the quaternary alloy under load as a function of compositions x and y is determined as:
aalloy ¼ xaInSb þ yaInP þ ð1 x yÞaInAS
ð4Þ
where aInSb, aInP and aInAs are the lattice constants of InSb, InP and InAs, respectively. The longitudinal optical (LO) and transversal optical (TO) phonon frequencies have been determined by combining the Lyddane–Sachs–Teller relation [13] with the expression [14]: 2
x2LO x2TO ¼
4p eT e2 M X0 e1
ð5Þ
where e1, eT , M and X0 are the high-frequency dielectric constant, the transverse effective charge, twice the reduced mass and the volume occupied by one atom, respectively. eT , e1 and e0 have been calculated using the same procedure as that used by Bouarissa et al. [15], where the refractive index is determined using Hervé and Vandamme relation [16], e1 is taken to be the square of the refractive index, and e0 is obtained within the Harrison model as reported in [14]. 3. Results and discussion
ð1Þ
where Vp(r) is the smoothly-varying pseudopotential which can be expanded into a Fourier series over the reciprocal lattice. Because the pseudopotential in a crystal lattice is periodic, it follows that the pseudo-wave function corresponding to (1) is also periodic. The valence electrons are assumed to be weakly bound to their host atoms so that the pseudo-wave function can be expanded into a Fourier series of plane waves. The EPM involves adjusting the pseudopotential form factors so as to reproduce measured band-gap energies at selected points in the Brillouin zone. The empirical pseudopotential parameters are optimized using the non-linear least-squares method [11,12]. The experimental band-gap energies at room temperature for InAs, InSb and InP at U, X and L high-symmetry points used in the fitting procedure are given in Table 1. The solution to the energy eigenvalues and corresponding eigenvectors can then be found by diagonalizing the Hamiltonian matrix. For a typical semiconductor system, 136 plane waves are sufficient, each corresponding to vectors in the reciprocal lattice, to expand the pseudopotential. The final local adjusted symmetric VS and antisymmetric VA pseudopotential form factors and the used lattice constants at room temperature for InAs, InSb and InP compounds are listed in Table 2. The formalism can be easily generalized to the case of an alloy using the VCA. In practice, VS and VA of an alloy can be expressed in terms of those of the binary compounds involved. Hence, for the
Table 1 Experimental band-gap energies at room temperature [3] for InAs, InSb and InP fixed in the fits. Compound
EC C (eV)
EXC (eV)
ELC (eV)
InAs InSb InP
0.36 0.18 1.35
1.37 1.63 2.21
1.07 0.93 2.05
The lattice matching condition for InAs1xySbxPy on the InAs substrate is as follows:
y ¼ 2:2857143x
ð0 6 x 6 0:30; 0 6 y 6 0:69Þ
ð6Þ
The dielectric constant is a measure of a substance’s ability to insulate charges from each other. Thus, it is a material’s ability to store a charge when used as a capacitor dielectric. In the present work, both dielectric constants, namely e1 and e0 have been calculated for different composition parameters x and y. Our results regarding e1 and e0 for InAs, InSb and InP are listed in Table 3. Also shown for comparison are the available data reported in the literature. Note that for both quantities e1 and e0, our results agree generally well with those reported in Refs. [13,17,18]. The variation of e1 and e0 as a function of composition x for InAs1xySbxPy lattice matched to InAs substrate is displayed in Fig. 1. We observe that as the Sb content increases from 0 to 0.30, e1 decreases from 14.08 to 11.12. The same behavior can be also seen for e0 which decreases from 15.41 to 13.73. The decrease of the dielectric constants by increasing the composition x means that the polarity becomes lower and the ability to stabilize charges for the material of interest becomes weaker. When the lattice matching condition given by Eq. (6) is not satisfied, for each couple (x, y) (0 6 x 6 0.30, 0 6 y 6 0.69), there is of course a difference between the lattice constant of the quaternary alloy of interest and that of the substrate (InAs in this work). The lattice mismatch Da/aInAs is expressed by:
ðaalloy aInAs Þ Da ¼ 100% aInAs aInAs
ð7Þ
where aInAs is the lattice constant of InAs given in Table 2. Fig. 2 illustrates the lattice mismatch dependence of e1 and e0 for InAs0.9ySb0.1Py. Note that as the lattice mismatch is varied on going from Da/aInAs = 2% to Da/aInAs = 0.7%, both parameters e1 and e0 are highly increased. This suggests the increase of the ability of the quaternary system under load to hold large quantities of charge for a long period of time.
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Table 2 Pseudopotential parameters for InAs, InSb and InP. Compound
Form factors (Ry)
InAs InSb InP
Lattice constant (Å)
VS(3)
VS(8)
VS(11)
VA(3)
VA(4)
VA(11)
0.217669 0.201294 0.213870
0.011 0.01 0.00
0.041633 0.028338 0.070502
0.054731 0.064495 0.088816
0.039485 0.03 0.06
0.024 0.015 0.03
Table 3 Optical high-frequency dielectric constant (e1) and optical static dielectric constant (e0) of InAs, InSb and InP.
16
15 Compound
e1
e0
InAs
14.08a; 11.6b 12.3c; 12.3d
15.42a; 14.3b 15.15c; 14.9d
14
InSb
15.43a; 15.3b 15.7c; 15.6d
17.99a; 17.2b 16.8c; 17.7d
13
a
InP a b c d
b
a
9.20 ; 9.9 9.61c; 9.6d
11.85 ; 12.9 12.5c; 12.4d
6.058 6.49 5.869
InAs0.9-y Sb0.1 Py
b
12
This work. Ref. [17]. Ref. [18]. Ref. [13].
11
10
High frequency dielectric constant Static dielectric constant
9
16
-2.0
InAs1-x-y Sbx Py /InAs
15
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 2. High-frequency and static dielectric constants in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
14 Table 4 Longitudinal optical (LO) and transversal optical (TO) phonon frequencies of InAs, InSb and InP.
13
Compound
12
11
10 0.00
InAs InSb InP
High frequency dielectric constant Static dielectric constant 0.05
0.10
0.15
0.20
a b
0.25
0.30
Composition X Fig. 1. High-frequency and static dielectric constants in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
Phonons play a major role in many of the physical properties of condensed matter, such as thermal conductivity and electrical conductivity. Their accurate description results in better understanding of structural parameters responsible for the efficiency of optical devices [19,20]. In the zinc-blende structures, such as the case here, one has to deal only with the transverse optical (TO) phonon frequency (xTO) and the longitudinal optical (LO) phonon frequency (xLO) at the U-point in the Brillouin zone. In the present contribution, xTO and xLO have been calculated at the zone-center for zinc-blende InAs1xySbxPy system. Our results for InAs, InSb and InP are given in Table 4. Also shown for comparison are the available experimental and theoretical data in the literature. The agreement between our results and data quoted in Ref. [13] seems to be reasonably good. The variation in the TO and LO phonon frequencies for zincblende InAs1xySbxPy lattice matched to InAs as a function of the composition x in the range 0–0.30 is shown in Fig. 3. We observe that by increasing Sb concentration x, both mode frequencies, i.e. xTO
xLO (1013 s1) a
b
5.20 ; 4.5 3.42a; 3.7b 7.1a; 6.5b
xTO (1013 s1) 4.97a; 4.1b 3.16a; 3.5b 6.25a; 5.7b
This work. Ref. [13].
and xLO shift to high frequencies. The shift is monotonic. Thus, larger values of xTO and xLO can be obtained at large compositions x. In Fig. 4, xTO and xLO are plotted against the lattice mismatch for InAs0.9ySb0.1Py. Note that by varying the lattice mismatch from 2% to 0.7%, both parameters of interest, namely xTO and xLO decrease monotonically and are shifted to low frequencies. The trend of the TO and LO frequencies with respect to x in Fig. 3 reflects the widening of the LO–TO splitting as one proceeds from x = 0 to x = 0.30. In this respect, the LOTO splitting in InAs1xySbxPy lattice matched to InAs has been plotted as a function of Sb content (x) in Fig. 5. An inspection of Fig. 5. shows that as the Sb concentration increases from 0 to 0.30, the LO–TO splitting increases as well from 0.23 1013 to 0.67 1013 s1. The increase is monotonic and almost linear. On the other hand, as can be seen from Fig. 4, the trend of the TO and LO frequencies with respect to the lattice mismatch reflects a the narrowing of the LO–TO splitting as one goes from aDInAs ¼ 2% a to aDInAs ¼ 0:7%. In this regard, the LO–TO splitting in InAs0.9ySb0.1Py is plotted versus the lattice mismatch in Fig. 6. We observe that as the lattice mismatch varies from 2% to 0.7%, the LO–TO splitting decreases monotonically and almost linearly from 0.78 1013 to 0.24 1013 s1. It is worth noting that for all compositions and lattice mismatching of interest for the quaternary system in question, the LO–TO splitting at the U high-symmetry point in
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6.25
6.00
InAs1-x-y Sbx Py /InAs 5.75
5.50
5.25
5.00
TO phonon frequency LO phonon frequency
4.75
4.50 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Composition X Fig. 3. Frequencies of the LO and TO phonons in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
7.0
TO phonon frequency LO phonon frequency
6.5
6.0
5.5
5.0
InAs0.9-y Sb0.1 Py 4.5 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 4. Frequencies of the LO and TO phonons in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
0.8
0.7
InAs1-x-y Sbx Py /InAs 0.6
0.5
0.4
0.3
0.2
0.1 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Composition X Fig. 5. LO–TO splitting in InAs1xySbxPy lattice matched to InAs versus the composition parameter x.
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0.9 0.8
InAs0.9-y Sb0.1 Py
0.7 0.6 0.5 0.4 0.3 0.2 0.1 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Lattice mismatch (%) Fig. 6. LO–TO splitting in InAs0.9ySb0.1Py versus the lattice mismatch percentage.
the Brillouin zone does not vanish. This could be due to the partial ionic bonding of the material system under load. 4. Conclusion In summary, the dielectric and vibrational properties of lattice matched and mismatched InAs1x-ySbxPy quaternary alloys have been investigated within the framework of the EPM under the VCA. The agreement between our findings and the available experimental and theoretical data reported in the literature is found to be reasonably good. The composition x dependence of dielectric constants, optical phonon frequencies and LO–TO splitting for InAs1xySbxPy quaternary system lattice matched to InAs has been studied showing a monotonic behavior of all parameters of interest versus x (0 6 x 6 0.30). An inspection of the variation of the static and high-frequency dielectric constants, LO and TO phonon frequencies and LO–TO splitting as a function of lattice mismatch percentage in InAs0.9ySb0.1Py showed that all quantities of interest are strongly dependent on the lattice mismatch percentage. The results obtained in the present contribution suggest that the variation of the alloy composition x and/or the lattice mismatch percentage of the quaternary system of interest may lead to more opportunities for obtaining diverse optical and lattice dynamical properties, which can provide a basis for understanding future device concepts and applications. Conflict of interest There is no conflict of interest. References [1] S. Adachi, Physical Properties of III–V Semiconductor Compounds, Wiley, New York, 1992.
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